Related work.
Kernelization for chordal completion goes back to the '90s and the seminal paper of Kaplan, Shamir and Tarjan [1]}. In this work, the
authors provide an \(O(k^3)\)  vertex-kernel for Chordal Completion (also known as Minimum Fill-In) that was later improved to \(O(k^2)\)  by a tighter analysis [2]}. Since then, several authors addressed completion, deletion and/or editing problems towards sub-classes of chordal graphs, as 3-leaf power graphs [3]}, split and threshold graphs [4]}, proper interval graphs [5]}, trivially perfect graphs [6]}, [7]}, [8]}, [4]} or ptolemaic graphs [10]}. All these classes have in common that they can be defined as chordal graphs, plus a constant number of obstructions. Several questions remain open, for example it is not known whether chordal deletion or chordal editing admit polynomial kernels [11]}.
